The movie Luke turns on in the hospital is The Philadelphia Story, a classic Katherine Hepburn/Cary Grant/Jimmy Stewart film. The storyline has parallels to Lorelai's relationship to Christopher. Hepburn and Grant play a rich society couple who grew up together and fell in love. The scene Luke is watching is just before they get married for the second time. When they mention 'eloping to Maryland', Luke realises why Lorelai mentioned Maryland when she asked him to elope.
While it does require certain steps be followed, marriages in France by US citizens are recognized in the US.
Rory talks about interviewing Senator Barack Obama and asking if he was going to run for president in 2008. Barack Obama was elected president in 2008.
When Logan is leaving the apartment so Lorelai and Rory can have a moment to talk alone, Luke (Matt) accidentally calls Lorelai by the actresses' name, Lauren.